Invariants
| Level: | $48$ | $\SL_2$-level: | $8$ | Newform level: | $2304$ | ||
| Index: | $48$ | $\PSL_2$-index: | $48$ | ||||
| Genus: | $2 = 1 + \frac{ 48 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 6 }{2}$ | ||||||
| Cusps: | $6$ (none of which are rational) | Cusp widths | $8^{6}$ | Cusp orbits | $2\cdot4$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | $1$ | ||||||
| $\Q$-gonality: | $2$ | ||||||
| $\overline{\Q}$-gonality: | $2$ | ||||||
| Rational cusps: | $0$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 8B2 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 48.48.2.95 |
Level structure
| $\GL_2(\Z/48\Z)$-generators: | $\begin{bmatrix}9&14\\22&23\end{bmatrix}$, $\begin{bmatrix}21&31\\2&7\end{bmatrix}$, $\begin{bmatrix}35&1\\32&17\end{bmatrix}$, $\begin{bmatrix}43&21\\36&13\end{bmatrix}$ |
| Contains $-I$: | yes |
| Quadratic refinements: | none in database |
| Cyclic 48-isogeny field degree: | $32$ |
| Cyclic 48-torsion field degree: | $512$ |
| Full 48-torsion field degree: | $24576$ |
Jacobian
| Conductor: | $2^{14}\cdot3^{4}$ |
| Simple: | no |
| Squarefree: | yes |
| Decomposition: | $1^{2}$ |
| Newforms: | 576.2.a.c, 2304.2.a.i |
Models
Embedded model Embedded model in $\mathbb{P}^{4}$
| $ 0 $ | $=$ | $ x y - z w $ |
| $=$ | $y^{2} + z^{2} - z t$ | |
| $=$ | $x z - x t + y w$ | |
| $=$ | $64 x^{2} - 3 y^{2} - 3 z t + 16 w^{2} - 3 t^{2}$ |
Singular plane model Singular plane model
| $ 0 $ | $=$ | $ x^{6} + 4 x^{4} z^{2} - 3 x^{2} y^{2} z^{2} + 2 x^{2} z^{4} - 12 y^{2} z^{4} $ |
Weierstrass model Weierstrass model
| $ y^{2} $ | $=$ | $ 3x^{6} + 24x^{4} + 54x^{2} + 24 $ |
Rational points
This modular curve has real points and $\Q_p$ points for $p$ not dividing the level, but no known rational points.
Maps between models of this curve
Birational map from embedded model to plane model:
| $\displaystyle X$ | $=$ | $\displaystyle y$ |
| $\displaystyle Y$ | $=$ | $\displaystyle \frac{4}{3}w$ |
| $\displaystyle Z$ | $=$ | $\displaystyle z$ |
Birational map from embedded model to Weierstrass model:
| $\displaystyle X$ | $=$ | $\displaystyle y^{2}$ |
| $\displaystyle Y$ | $=$ | $\displaystyle 4y^{4}zw+16y^{2}z^{3}w$ |
| $\displaystyle Z$ | $=$ | $\displaystyle yz$ |
Maps to other modular curves
$j$-invariant map of degree 48 from the embedded model of this modular curve to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle 2^6\cdot3\,\frac{36859392z^{2}w^{6}+119386944z^{2}w^{4}t^{2}+36381096z^{2}w^{2}t^{4}+1879335z^{2}t^{6}+348994560zw^{6}t+179978112zw^{4}t^{3}-795888zw^{2}t^{5}-1104462zt^{7}+73728w^{8}+112076288w^{6}t^{2}+40847424w^{4}t^{4}-7475832w^{2}t^{6}-773415t^{8}}{130560z^{2}w^{6}-10944z^{2}w^{4}t^{2}+1512z^{2}w^{2}t^{4}-81z^{2}t^{6}+89088zw^{6}t-1152zw^{4}t^{3}-2160zw^{2}t^{5}+162zt^{7}+8192w^{8}+13824w^{6}t^{2}+576w^{4}t^{4}-1080w^{2}t^{6}+81t^{8}}$ |
Modular covers
Cover information
Click on a modular curve in the diagram to see information about it.
|
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 16.24.0.l.1 | $16$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 24.24.1.dk.1 | $24$ | $2$ | $2$ | $1$ | $1$ | $1$ |
| 48.24.1.f.1 | $48$ | $2$ | $2$ | $1$ | $0$ | $1$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 48.96.3.v.1 | $48$ | $2$ | $2$ | $3$ | $1$ | $1$ |
| 48.96.3.dj.1 | $48$ | $2$ | $2$ | $3$ | $1$ | $1$ |
| 48.96.3.ik.1 | $48$ | $2$ | $2$ | $3$ | $2$ | $1$ |
| 48.96.3.iq.1 | $48$ | $2$ | $2$ | $3$ | $2$ | $1$ |
| 48.96.3.rz.1 | $48$ | $2$ | $2$ | $3$ | $2$ | $1$ |
| 48.96.3.se.1 | $48$ | $2$ | $2$ | $3$ | $2$ | $1$ |
| 48.96.3.uy.1 | $48$ | $2$ | $2$ | $3$ | $1$ | $1$ |
| 48.96.3.vd.1 | $48$ | $2$ | $2$ | $3$ | $1$ | $1$ |
| 48.144.10.lq.1 | $48$ | $3$ | $3$ | $10$ | $4$ | $1^{4}\cdot2^{2}$ |
| 48.192.11.gi.1 | $48$ | $4$ | $4$ | $11$ | $3$ | $1^{7}\cdot2$ |
| 240.96.3.cyv.1 | $240$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 240.96.3.czd.1 | $240$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 240.96.3.dbh.1 | $240$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 240.96.3.dbp.1 | $240$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 240.96.3.dsn.1 | $240$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 240.96.3.dsv.1 | $240$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 240.96.3.duz.1 | $240$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 240.96.3.dvh.1 | $240$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 240.240.18.iq.1 | $240$ | $5$ | $5$ | $18$ | $?$ | not computed |
| 240.288.19.wkr.1 | $240$ | $6$ | $6$ | $19$ | $?$ | not computed |